The Two Secant Angle Theorem is included in the 2016 Alabama Course of Study under “Understand and apply theorems about circles” (Alabama Department of Education, 2016). Students should be able to identify central angles, inscribed angles, diameters, radii chords, tangents, and secants.

25. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. [G-C2]

26. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [G-C3]

 This theorem would be beneficial to explore with high school Geometry students because the Alabama Course of Study standards above are included in the geometry portion of the document. Not only does this theorem involve geometry, but the additional questions below include algebraic computations as well.

Students can explore this theorem at Secant Lines Intersecting a Circle – Arcs and Angles. I would have them begin by moving the points A, C, D, and E, and then take notes on how moving those points affect angle B. I want to see if they notice the angle is ½ of the difference of the two arcs on their own. Next, I would follow up by asking questions such as:

1. If m arc AC=210 degrees and m arc DE=70 degree, find m∠B.
2. If m arc AC=120 degree and m∠B=15 degrees, find m arc DE.
3. If m∠B=10 degrees and m arc DE=40 degrees, find m arc AC.

Instead of giving the formula to students right off the bat, I want them to try and discover it for themselves by using their prior knowledge about the Inscribed Angle Theorem and the Exterior Angle Theorem. I would put students in groups of two-three students to discuss their findings and to make observations. According to Principles to Actions,  “The role of the teacher is to engage students in tasks that promote reasoning and problem solving and facilitate discourse that moves students toward shared understanding or mathematics” (National Council of Teachers of Mathematics, 2014, pp. 11).  Therefore, having students work in groups will allow them to facilitate discourse while they problem solve and reason about this theorem.

Continue on to the Conclusion.

Revisit the Theorem Solution if there is still any confusion.